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lemma isin_sorted_split_list_right:
assumes "split_list ts x = (ls, sep#rs)"
and "sorted_less ts"
shows "x \<in> set (sep#rs) = (x = sep)" |
lemma hnorm_triangle_ineq: "\<And>x y::'a::real_normed_vector star. hnorm (x + y) \<le> hnorm x + hnorm y" |
lemma std_normal_moment_even:
"has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k)) (fact (2 * k) / (2^k * fact k))" |
lemma has_derivative_within_alt:
"(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x))" |
lemma cnj_of_int [simp]: "cnj (of_int z) = of_int z" |
lemma fv_eqvt[simp,eqvt]: "\<pi> \<bullet> (fv e) = fv (\<pi> \<bullet> e)" |
lemma exp_Lambert_W': "x \<in> {-exp (-1)..<0} \<Longrightarrow> exp (Lambert_W' x) = x / Lambert_W' x" |
lemma sum_atLeast_Suc_shift: "0 < b \<Longrightarrow> a \<le> b \<Longrightarrow> sum f {Suc a..b} = (\<Sum>i=a..b - 1. f (Suc i))" |
lemma Contra: "insert (neg A) H \<turnstile> A \<Longrightarrow> H \<turnstile> A" |
lemma PPs\<alpha>_of_step:
"\<langle>c,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>
\<Longrightarrow> set (PPV \<alpha>) \<subseteq> set (PPc c)" |
lemma kp_5x9_ur_last: "last kp5x9ur = (4,8)" |
lemma delay_lazy_cong: "delay f = delay f" |
lemma PK_imply_weaken:
assumes \<open>A; B \<turnstile>\<^sub>! ps \<^bold>\<leadsto>\<^sub>! q\<close> \<open>set ps \<subseteq> set ps'\<close>
shows \<open>A; B \<turnstile>\<^sub>! ps' \<^bold>\<leadsto>\<^sub>! q\<close> |
lemma exec_strip_guards_to_exec:
assumes exec_strip: "\<Gamma>\<turnstile>\<langle>strip_guards F c,s\<rangle> \<Rightarrow> t"
shows "\<exists>t'. \<Gamma>\<turnstile>\<langle>c,s\<rangle> \<Rightarrow> t' \<and>
(isFault t \<longrightarrow> isFault t') \<and>
(t' \<in> Fault ` (-F) \<longrightarrow> t'=t) \<and>
(\<not> isFault t' \<longrightarrow> t'=t)" |
lemma Abs_swap_fresh:
assumes good_X: "good X" and fresh: "fresh xs x' X"
shows "Abs xs x X = Abs xs x' (X #[x' \<and> x]_xs)" |
lemma leftderives_induct[consumes 1, case_names Base Step]:
assumes derives: "leftderives a b"
assumes Pa: "P a"
assumes induct: "\<And>y z. leftderives a y \<Longrightarrow> leftderives1 y z \<Longrightarrow> P y \<Longrightarrow> P z"
shows "P b" |
lemma ordinal_wf: "wf {(x,y::ordinal). x < y}" |
lemma Transset_TC: "Transset(TC a)" |
lemma While_inter_right:
"interfree_aux_right \<Gamma> \<Theta> F (q, c, a)
\<Longrightarrow> interfree_aux_right \<Gamma> \<Theta> F (q, While b c, AnnWhile r i a)" |
lemma matches_rule_and_simp:
assumes "matches \<gamma> m p"
shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule (MatchAnd m m') a'], s\<rangle> \<Rightarrow> t \<longleftrightarrow> \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m' a'], s\<rangle> \<Rightarrow> t" |
lemma OclOr2[simp]: "(invalid or false) = invalid" |
lemma equivp_dlist_eq: "equivp dlist_eq" |
lemma interpretation_grounds_all':
"interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<Longrightarrow> ground (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)" |
lemma guha_umstaendlich: (* or maybe it's Coq where the original formulation is more beneficial *)
assumes ae: "a = ofe_action fe"
assumes ele: "fe \<in> set ft"
assumes rest: "\<gamma> (ofe_fields fe) p"
"\<forall>fe' \<in> set ft. ofe_prio fe' > ofe_prio fe \<longrightarrow> \<not>\<gamma> (ofe_fields fe') p"
shows "guha_table_semantics \<gamma> ft p (Some a)" |
lemma product_dvd_irreducibleD:
fixes a b x :: "'a :: algebraic_semidom"
assumes "irreducible x"
assumes "a * b dvd x"
shows "a dvd 1 \<or> b dvd 1" |
lemma singular_boundary:
"singular_relboundary p X {} c \<longleftrightarrow>
(\<exists>d. singular_chain (Suc p) X d \<and> chain_boundary (Suc p) d = c)" |
lemma simple_list_index_equality:
assumes "length a = n"
assumes "length b = n"
assumes "\<forall>i < n. a!i = b!i"
shows "a = b" |
lemma sizeof_nonzero:
"|t|\<^sub>\<tau> > 0" |
lemma Arr_in_Hom:
assumes "Arr t"
shows "t \<in> HHom (Src t) (Trg t)" and "t \<in> VHom (Dom t) (Cod t)" |
lemma Nil_in_Shuffle[simp]: "[] \<in> A \<parallel> B \<longleftrightarrow> [] \<in> A \<and> [] \<in> B" |
lemma zip_fun_eq[dest]:
assumes "f \<parallel> g = h \<parallel> i"
shows "f = h" "g = i" |
lemma a_consistent_thunk_once:
assumes "a_consistent (ae, a, as) (\<Gamma>, Var x, S)"
assumes "map_of \<Gamma> x = Some e"
assumes [simp]: "ae x = up\<cdot>u"
assumes "heap_upds_ok (\<Gamma>, S)"
shows "a_consistent (env_delete x ae, u, as) (delete x \<Gamma>, e, S)" |
lemma case_nat_in_state_measure[intro]:
assumes "x \<in> type_universe t1" "\<sigma> \<in> space (state_measure V \<Gamma>)"
shows "case_nat x \<sigma> \<in> space (state_measure (shift_var_set V) (case_nat t1 \<Gamma>))" |
lemma (in is_functor) ntcf_arrow_id_components:
"(ntcf_arrow_id \<AA> \<BB> (cf_map \<FF>))\<lparr>NTMap\<rparr> = ntcf_id \<FF>\<lparr>NTMap\<rparr>"
"(ntcf_arrow_id \<AA> \<BB> (cf_map \<FF>))\<lparr>NTDom\<rparr> = cf_map (ntcf_id \<FF>\<lparr>NTDom\<rparr>)"
"(ntcf_arrow_id \<AA> \<BB> (cf_map \<FF>))\<lparr>NTCod\<rparr> = cf_map (ntcf_id \<FF>\<lparr>NTCod\<rparr>)" |
lemma red_simps [simp]:
assumes "Ide t"
shows "Src (t\<^bold>\<down>) = Src t" and "Trg (t\<^bold>\<down>) = Trg t"
and "Dom (t\<^bold>\<down>) = t" and "Cod (t\<^bold>\<down>) = \<^bold>\<lfloor>t\<^bold>\<rfloor>" |
lemma raw_ports_compress_dst_CannotMatch:
fixes p :: "('i::len, 'a) tagged_packet_scheme"
assumes generic: "primitive_matcher_generic \<beta>"
and c: "l4_ports_compress pss = CannotMatch"
shows "\<not> matches (\<beta>, \<alpha>) (alist_and (map (Pos \<circ> Dst_Ports) pss)) a p" |
lemma r_swap_recfn [simp]: "recfn 2 f \<Longrightarrow> recfn 2 (r_swap f)" |
lemma wf_chains2:
fixes k
assumes "S |\<subseteq>| initials G"
and "wf_dia G"
and "\<Pi> \<in> ps_chains2 S G"
and "fcard G^V + length G^E = k + fcard S"
shows "wf_ps_chain \<Pi> \<and> (post \<Pi> = [ terminals G |=> Bot ])" |
lemma exists_finite_inconsistent:
assumes \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> V)\<close>
obtains W where \<open>{\<^bold>\<not> p} \<union> W \<subseteq> {\<^bold>\<not> p} \<union> V\<close> \<open>(\<^bold>\<not> p) \<notin> W\<close> \<open>finite W\<close> \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> W)\<close> |
lemma msig_qmdecl_simp[simp]: "msig (qmdecl sig m) = sig" |
lemma val_ringI:
assumes "a \<in> carrier Q\<^sub>p"
assumes "val a \<ge>0"
shows " a \<in> \<O>\<^sub>p" |
lemma add_anno_rel_GuarNoWrite [simp]:
"\<Gamma> \<oplus>\<^sub>\<S> (v -=\<^sub>m GuarNoWrite) = \<Gamma>" |
lemma ground_head: "ground s \<Longrightarrow> is_Sym (head s)" |
lemma quadratic_linear1:
assumes "b\<noteq>0"
assumes "a \<noteq> 0"
assumes "4 * a * ba \<le> aa\<^sup>2"
assumes "(b::real) * (sqrt ((aa::real)\<^sup>2 - 4 * (a::real) * (ba::real)) - (aa::real)) / (2 * a) + (c::real) = 0"
assumes "
(\<forall>x\<in>set (les::(real*real*real)list).
case x of
(d, e, f) \<Rightarrow>
d * ((sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a))\<^sup>2 +
e * (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a) +
f
< 0)"
assumes "(aaa, aaaa, baa) \<in> set les"
shows "aaa * (c / b)\<^sup>2 - aaaa * c / b + baa < 0" |
lemma obtain_fresh_z_c_of:
fixes t::"'b::fs"
obtains z where "atom z \<sharp> t \<and> \<tau> = \<lbrace> z : b_of \<tau> | c_of \<tau> z \<rbrace>" |
lemma (in comm_group) rel_in_carr:
assumes "A \<subseteq> carrier G" "r \<in> relations A"
shows "(\<lambda>a. a [^] r a) \<in> A \<rightarrow> carrier G" |
lemma rbt_insert_entries_Some: "is_rbt t \<Longrightarrow> rbt_lookup t k = Some v' \<Longrightarrow>
set (RBT_Impl.entries (rbt_insert k v t)) = insert (k, v) (set (RBT_Impl.entries t) - {(k, v')})" |
lemma src_ipPart_motivation:
fixes rs
defines "X \<equiv> (\<lambda>(base,len). ipset_from_cidr base len) ` src ` match_sel ` set rs"
assumes "\<forall>A \<in> X. B \<subseteq> A \<or> B \<inter> A = {}" and "s1 \<in> B" and "s2 \<in> B"
shows "simple_fw rs (p\<lparr>p_src:=s1\<rparr>) = simple_fw rs (p\<lparr>p_src:=s2\<rparr>)" |
lemma mkKripke_simps[simp]:
"worlds (mkKripke ws rels val) = ws"
"relations (mkKripke ws rels val) = (\<lambda>a. rels a \<inter> ws \<times> ws)"
"valuation (mkKripke ws rels val) = val" |
lemma sdrop_sconst: "sdrop n s = sconst x \<Longrightarrow> n \<le> m \<Longrightarrow> s !! m = x" |
lemma bijection_transpose:
\<open>bijection (transpose a b)\<close> |
lemma reach2:
"\<not> s\<turnstile> l reachable_from x \<Longrightarrow> \<not> s\<langle>l:=y\<rangle>\<turnstile> l reachable_from x" |
lemma cdi_iff_no_strict_pd: \<open>i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<longleftrightarrow> is_path \<pi> \<and> k < i \<and> \<pi> i \<noteq> return \<and> (\<forall> j \<in> {k..i}. \<not> (\<pi> k, \<pi> j) \<in> pdt)\<close> |
lemma in_Chains_subset: "\<lbrakk> M \<in> Chains r; M' \<subseteq> M \<rbrakk> \<Longrightarrow> M' \<in> Chains r" |
lemma sync_jview_jAction_eq:
assumes traces: "{ t \<in> T . tLength t = n } = { t \<in> T' . tLength t = n }"
assumes tT: "t \<in> { t \<in> T . tLength t = n }"
shows "jAction (mkM T) t = jAction (mkM T') t" |
lemma zeta_partial_sum_le_neg:
assumes "s > 0"
shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>n. n powr s) x - x powr (1 + s) / (1 + s)\<bar> \<le> c * x powr s" |
lemma Zn_neq1_cyclic_group:
assumes "n \<noteq> 1"
shows "cyclic_group (Z n) 1" |
lemma addBalance_eq:
assumes "addBalance ad val acc = Some acc'"
and "ad \<noteq> ad'"
shows "(accessBalance acc ad') = (accessBalance acc' ad')" |
lemma Stirling_same [simp]: "Stirling n n = 1" |
theorem rinss_random_bst:
"distr (rinss xs \<langle>\<rangle> A) (tree_sigma (count_space A)) (map_tree fst) =
restrict_space (measure_pmf (random_bst (set xs))) (trees A)" |
lemma PosOrd_Stars_appendI:
assumes "Stars vs1 :\<sqsubset>val Stars vs2" "flat (Stars vs1) = flat (Stars vs2)"
shows "Stars (vs @ vs1) :\<sqsubset>val Stars (vs @ vs2)" |
lemma decr_sorted:
assumes "decr xs"
shows "sorted (rev xs)" |
lemma min_dist_z[simp]: "min_dist v v = 0" |
lemma empty_text[simp]: "get_text empty = []" |
lemma noWhileL_intro[intro]:
assumes "\<And> c. c \<in> set cl \<Longrightarrow> noWhile c"
shows "noWhileL cl" |
lemma Derives1_rule: "Derives1 a i r b \<Longrightarrow> r \<in> \<RR>" |
lemma "test (do {
doc \<leftarrow> return Node_removeChild_document;
s \<leftarrow> doc . createElement(''div'');
tmp0 \<leftarrow> s . ownerDocument;
assert_equals(tmp0, doc);
tmp1 \<leftarrow> Node_removeChild_document . body;
assert_throws(NotFoundError, tmp1 . removeChild(s));
tmp2 \<leftarrow> s . ownerDocument;
assert_equals(tmp2, doc)
}) Node_removeChild_heap" |
lemma i_Exec_Comp_Stream_Acc_Output__eq_Msg_State_Idle_conv': "
\<lbrakk> Suc 0 < k;
State_Idle localState output_fun trans_fun (
i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t);
m \<noteq> \<NoMsg>;
t0 = t * k;
s = i_Exec_Comp_Stream trans_fun (input \<odot>\<^sub>i k) c \<rbrakk> \<Longrightarrow>
(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m) =
(((\<not> State_Idle localState output_fun trans_fun (localState (s t2))). t2 \<U> t1 [0\<dots>] \<oplus> t0.
(output_fun (s t1) = m \<and> State_Idle localState output_fun trans_fun (localState (s t1)))) \<or>
((\<not> State_Idle localState output_fun trans_fun (localState (s t2))). t2 \<U> t1 [0\<dots>] \<oplus> t0.
(output_fun (s t1) = m \<and>
(\<circle> t3 t1 [0\<dots>].
((output_fun (s t4) = \<NoMsg>. t4 \<U> t5 ([0\<dots>] \<oplus> t3).
(output_fun (s t5) = \<NoMsg> \<and> State_Idle localState output_fun trans_fun (localState (s t5)))))))))" |
lemma summable_on_cinner_left:
assumes \<open>f summable_on I\<close>
shows \<open>(\<lambda>i. cinner a (f i)) summable_on I\<close> |
lemma Pring_zero:
"\<zero>\<^bsub>Pring R I\<^esub> = indexed_const \<zero>" |
lemma region_addresses_iff: "a' \<in> region_addresses a si \<longleftrightarrow> unat (a' - a) < si" |
lemma vwalk_arcs_tl_empty:
"vwalk_arcs xs = [] \<Longrightarrow> vwalk_arcs (tl xs) = []" |
lemma DERIV_continuous_on:
"(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x within s)) \<Longrightarrow> continuous_on s f" |
lemma stream_numeral [simp]: "stream (numeral n) = numeral n" |
lemma set_of_mul_inc_right:
"set_of (A * B) \<subseteq> set_of (A * B')"
if "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_ring interval" |
lemma bounded_clinear_CBlinfun_apply: "bounded_clinear f \<Longrightarrow> cblinfun_apply (CBlinfun f) = f" |
lemma max_triangle_bound: "triangle z \<le> e \<Longrightarrow> z \<le> e" |
lemma sign_changes_0_Cons [simp]:
"sign_changes (0 # xs :: 'a :: idom_abs_sgn list) = sign_changes xs" |
lemma fmrestrict_set_inter_img:
fixes A X Y
shows "fmrestrict_set (X \<inter> Y) ` A = (fmrestrict_set X \<circ> fmrestrict_set Y) ` A" |
lemma order_le_prod [iff]:
"order(Product.le rA rB) = (order rA & order rB)" |
lemma b_assn_invalid_merge3: "hn_invalid A x y \<or>\<^sub>A hn_invalid (b_assn A P) x y
\<Longrightarrow>\<^sub>t hn_invalid A x y" |
lemma (in \<Z>) smc_CAT_obj_initialD:
assumes "obj_initial (smc_CAT \<alpha>) \<AA>"
shows "\<AA> = cat_0" |
lemma condensation_condense1: "(\<Sum>k=1..<2^n. f (2 ^ Discrete.log k)) = (\<Sum>k<n. 2^k * f (2 ^ k))" |
lemma plossless_fail_converter [simp]: "plossless_converter \<I> \<I>' fail_converter \<longleftrightarrow> \<I> = bot" (is "?lhs \<longleftrightarrow> ?rhs") |
lemma ord_of_minus_1: "n > 0 \<Longrightarrow> ord_of n = succ (ord_of (n - 1))" |
lemma fpairs_vinsert: "fpairs (vinsert [a, b]\<^sub>\<circ> A) = set {[a, b]\<^sub>\<circ>} \<union>\<^sub>\<circ> fpairs A" |
lemma Abs_sat'_eq_of_nat: "Abs_sat' n = of_nat n" |
lemma l3_lkr_after_refines_lkr_after:
"{R23s} l2_lkr_after A, l3_lkr_after A {>R23s}" |
lemma sort_rev_eq_sort: "distinct xs \<Longrightarrow> sort (rev xs) = sort xs" |
lemma is_weak_order_B: "weak_partial_order \<Y>" |
lemma "!!a::real. a + b + c + d \<le> i + j + k + l ==> a \<le> b ==> b \<le> c
==> c \<le> d ==> i \<le> j ==> j \<le> k ==> k \<le> l ==> a + a + a + a + a + a \<le> l + l + l + l + i + l" |
lemma [def_pat_rules]:
"Network.resCap_cf_impl$c \<equiv> UNPROTECT resCap_cf_impl" |
lemma fresh_star_insert_elim:
"(insert x S \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (x \<sharp> c \<Longrightarrow> S \<sharp>* c \<Longrightarrow> PROP C)" |
lemma N_vector_top_pp [simp]:
"N (x * top) * top = --(x * top)" |
lemma vcompI[intro!]:
assumes "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s"
shows "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s" |
lemma alphaAbs_qAbs_iff_alphaAbs_all_equal_or_qFresh:
assumes "qGood X" and "qGood X'"
shows "(qAbs xs x X $= qAbs xs' x' X') =
alphaAbs_all_equal_or_qFresh xs x X xs' x' X'" |
lemma (in CRR_market) case_asset:
assumes "asset \<in> stocks Mkt"
shows "asset = stk \<or> asset = risk_free_asset" |
lemma alprio_delete_correct:
assumes "al_annot \<alpha> invar annot"
and "al_splits \<alpha> invar splits"
and "al_app \<alpha> invar app"
shows "prio_delete (alprio_\<alpha> \<alpha>) (alprio_invar \<alpha> invar)
(alprio_find annot) (alprio_delete splits annot app)" |
lemma map_key_PP_sum: "Poly_Mapping.map_key PP (sum f A) = (\<Sum>a\<in>A. Poly_Mapping.map_key PP (f a))" |
lemma act_morph2: "\<alpha> 1 = id" |
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